Renewable Energy 41 (2012) 105e114 Contents lists available at SciVerse ScienceDirect Renewable Energy journal homepage: www.elsevier.com/locate/renene Hydrodynamics of the IPS buoy wave energy converter including the effect of non-uniform acceleration tube cross section António F.O. Falcão a,*, José J. Cândido a, Paulo A.P. Justino b, João C.C. Henriques a a IDMEC, Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal b Laboratório Nacional de Energia e Geologia, Estrada Paço do Lumiar, 1649-038 Lisbon, Portugal article info Article history: Received 15 June 2011 Accepted 6 October 2011 Available online 6 November 2011 Keywords: Wave energy Wave power IPS buoy Oscillating body Hydrodynamics abstract An important class of ﬂoating wave energy converters (that includes the IPS buoy, the Wavebob and the PowerBuoy) comprehends devices in which the energy is converted from the relative (essentially heaving) motion between two bodies oscillating differently. The paper considers the case of the IPS buoy, consisting of a ﬂoater rigidly connected to a fully submerged vertical (acceleration) tube open at both ends. The tube contains a piston whose motion relative to the ﬂoater-tube system (motion originated by wave action on the ﬂoater and by the inertia of the water enclosed in the tube) drives a power take-off mechanism (PTO) (assumed to be a linear damper). To solve the problem of the end-stops, the central part of the tube, along which the piston slides, bells out at both ends to limit the stroke of the piston. The use of a hydraulic turbine inside the tube is examined as an alternative to the piston. A frequency domain analysis of the device in regular waves is developed, combined with a one-dimensional unsteady ﬂow model inside the tube (whose cross section is in general non-uniform). Numerical results in regular and irregular waves are presented for a cylindrical buoy with a conical bottom, including the optimization of the acceleration tube geometry and PTO damping coefﬁcient for several wave periods. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction The concept of the point absorber for wave energy utilization was developed in the late 1970s and early 1980s, mostly in Scandinavia [1]. This is in general a wave energy converter of oscillating body type whose horizontal dimensions are small compared to the representative wavelength. In its simplest version, the body reacts against the bottom. In deep water (say 50 m or more), this may raise difﬁculties due to the distance between the ﬂoating body and the sea bottom. Multi-body systems may then be used instead, in which the energy is converted from the relative motion between two bodies oscillating differently. This is the case of several devices presently under development, like the Pelamis, the Wavebob and the PowerBuoy. Sometimes the relevant relative motion results from heaving oscillations. This paper considers the special situation when a ﬂoater reacts against the inertia of the water contained in a long vertical tube open at both ends, located underneath. This is the case of the spar-buoy OWC, possibly the simplest concept for a ﬂoating oscillating water column (OWC) device equipped with an air turbine, in which the upper end of the tube extends through the * Corresponding author. E-mail addresses: falcao@hidrol.ist.utl.pt, antonio.falcao@ist.utl.pt (A.F.O. Falcão). 0960-1481/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2011.10.005 buoy above the sea water level. Masuda, in Japan, developed a navigation buoy based on the OWC spar-buoy concept [2,3]. The spar-buoy OWC will not be analysed in this paper. The IPS buoy is another type of spar-buoy and will be analysed here in detail. It was invented by Noren [4] and initially developed in Sweden by the company Interproject Service (IPS). The device consists of a buoy rigidly connected to a fully submerged vertical tube (the so-called acceleration tube) open at both ends (Fig. 1). The tube contains a piston whose motion relative to the ﬂoater-tube system (motion originated by wave action on the ﬂoater and by the inertia of the water enclosed in the tube) drives a power takeoff (PTO) mechanism. The same inventor later introduced an improvement that signiﬁcantly contributes to solve the problem of the end-stops: the central part of the tube, along which the piston slides, bells out at both ends to limit the stroke of the piston [5]. A half-scale prototype of the IPS buoy was tested in sea trials in Sweden, in the early 1980s [6]. The AquaBuOY is a wave energy converter, developed in the 2000s, that combines the IPS buoy concept with a pair of hose pumps to produce a ﬂow of water at high pressure that drives a Pelton turbine [7,8]. A prototype of the AquaBuOY was deployed and tested in 2007 in the Paciﬁc Ocean off the coast of Oregon. A variant of the initial IPS buoy concept, due to Stephen Salter, is the sloped IPS buoy: the natural frequency of the converter may be 106 A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114 analysis using tools like the commercially available codes (e.g. WAMIT, AQUADYN) based on the boundary element method for the computation of the hydrodynamic coefﬁcients, including the interference between the buoy and the acceleration tube, as done in [12]. 2.1. One-dimensional ﬂow inside the tube piston Fig. 1. Schematic representation of the IPS buoy. reduced, and in this way the capture width enlarged, if the buoytube set is made to oscillate at an angle intermediate between the heave and the surge directions. The sloped IPS buoy has been studied since the mid-1990s at the University of Edinburgh, by model testing and numerical modelling [9e11]. 2. Theoretical modelling The IPS buoy consists basically of a buoy rigidly connected to a fully submerged tube (the acceleration tube), oscillating in heave, by the action of the waves, with respect to a piston that can slide along the tube. The wave energy is absorbed by means of the relative motion between the piston and the buoy-tube set. The concept is represented in Fig. 1. We note that most of the inertia against which the buoy-tube set moves is that of the water contained inside the acceleration tube (obviously in addition to the mass of the piston itself). In the simpliﬁed mathematical modelling adopted in this paper, we assume that the buoy-tube set is constrained to oscillate in heave, an assumption that seems reasonable taking into account the axial extent of the device. We introduce the following assumptions. (i) The tube is sufﬁciently far away underneath the buoy for the hydrodynamic interaction between both to be negligible. (ii) The interaction between the wave ﬁelds induced by the two ends of the tube may be neglected. (iii) The distance from the free surface to the tube upper end is large enough for the excitation and radiation forces on the ﬂow about the two tube ends to be neglected. (We note however that the added mass at the two ends of the tube will be accounted for.) (iv) Finally, the ﬂow inside the tube is modelled as onedimensional. Admittedly, some of these simpliﬁcations may be rather drastic. This is specially the case of assumptions (i) and (iii) if the distances from the acceleration tube to the buoy and to the free surface are not large enough. In spite of this, the present paper is expected to provide useful insights into the relationships between device geometry, PTO parameters and wave energy converter performance. Naturally, in cases of special practical interest, this simpliﬁed approach should be complemented by a more rigorous We consider now the ﬂow inside the acceleration tube, whose total length is L (Fig. 2). The position of the tube sections are deﬁned by a longitudinal coordinate x (with x ¼ 0 at the lower end of the tube). The piston is allowed to move, relative to the tube, inside a central part 3e4 (working part), b2 þ b3 x b1 þ b2 þ b3, of length b1 and cross sectional area A1, as shown in Fig. 2. The working part is continued downwards and upwards by tube parts 1e2 and 5e6, of lengths b3 and b4, respectively, both of cross sectional area A2 ¼ a2A1 (a ! 1). The transitions are provided by conical connections 2e3 and 4e5, of cross sectional areas A(x). If a > 1, there may be a signiﬁcant axial force on the tube resulting from the pressure distribution on the inner conical walls. The added mass of the oscillating water column contained in a semi-inﬁnite open tube of radius r and of negligible wall thickness in an unbounded perfect ﬂuid is r p l r2, where l ¼ 0.6133 r is an added length (see [13,14]). We assume this result to apply to our case (small tube thickness close to the tube ends), with p r2 ¼ A2. The introduction of the added mass (or the added length l) in a theoretical model that assumes the ﬂow inside the tube to be onedimensional allows the following two effects to be accounted for: (i) the inner ﬂow close to the tube end is non-uniform; (ii) the outer ﬂow ﬁeld in the vicinity of the tube end is affected by the motion of the water column inside the tube. These effects were analysed in two- or three-dimensional potential ﬂow in [13,14]. In a ﬁxed frame of reference, the buoy-tube pair moves along its own vertical axis with velocity W(t) (positive for upward motion), where t is time. We note that the water ﬂow inside the tube is unsteady in any referential. In our analysis, we adopt a non-inertial 6 b4 A2 5 b2 A(ξ ) 4 L b1 3 b2 2 b3 1 A1 A(ξ ) ξ A2 Fig. 2. IPS buoy with acceleration tube. A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114 107 frame of reference ﬁxed to the buoy-tube pair. In this referential, the piston velocity is V(t) and the one-dimensional ﬂow velocity at a section x of area A(x) is The force on the piston is fp(t) ¼ A1(p(zÀ,t) À p(zþ,t)), where zÀ and zþ are the coordinates of the lower and upper surfaces of the piston. We ﬁnd vðx; tÞ ¼ AAð1xÞVðtÞ: (1) For a conical transition, it is AðxÞ A1 ¼ ða À 1Þb3bÀ2 x þ 2 a ðb3 x b2 þ b3Þ; (2) AðxÞ A1 ¼ ða À x 1Þ À b1 À b2 b2 À b3 2 þ1 (3) ðb1 þ b2 þ b3 x b1 þ 2b2 þ b3Þ: In what follows, we assume that the piston is of negligible length and mass (this is equivalent to assuming that its length is non-zero and its mean density is equal to water density). Since the tube is totally submerged, the net force on the piston is not affected by gravity and so we simply ignore the acceleration of gravity for its calculation and denote by pout the uniform pressure of the supposedly unbounded water far away from the tube ends. Let x ¼ z be the instantaneous position of the piston (assumed of negligible length). Applying Bernoulli’s equation for unsteady onedimensional ﬂow (see e.g. [15]), we ﬁnd, for the pressure at a section x < z below the piston, " # pðx; tÞ ¼ pout þ r aÀ4 À A21 AðxÞ2 V2 2 À rðx þ lÞddWt À r Zx vvðx; tÞ vt dx: (4) Àl The second term on the right-hand-side of Eq. (4) accounts for the difference in kinetic energy at cross sections with different areas, and is zero where the cross sectional area is A2. The third term on the right-hand-side of Eq. (4), proportional to dW/dt, is due to the ﬁctitious body force per unit mass ÀdW/dt associated with the noninertial frame of reference. The last term results from the unsteadiness of the velocity v(x,t) and may be written as Àr Zx vvðx; tÞ vt dx Àl ¼ rddVt l þ b3 a2 þ aðab2 b2ðx À b3Þ À b3 þ ab3 þ x À axÞ ðb3 x b2 þ b3Þ; (5) Àr Zx vvðx;tÞdx vt ¼ rdV dt lþb3 a2 þba2 þxÀb2 Àb3 ðb2 þb3 x zÞ: (6) Àl We note that Eq. (4) gives p(Àl,t) ¼ pout, as if the tube were extended downwards by a length equal to the added length l. Above the piston, x > z, we have ! pðx; tÞ ¼ pout þ r aÀ4 À A21 AðxÞ2 V2 2 þ rðL À x þ lÞddWt þ r ZLþl vvðx; vt tÞ dx: (7) x Expressions similar to (5) and (6) can be derived for the last term of Eq. (7). fpðtÞ ¼ ÀMW dW dt À MV dV dt ; where MW ¼ rA1ðL þ 2lÞ; MV ¼ rA1 b1 þ aÀ2ðb3 þ b4 þ 2lÞ þ 2b2aÀ1 : (8) (9) (10) The total axial force ft(t) on the internal surface of the two conical parts of the acceleration tube is ftðtÞ ¼ bZ2þb3 pðx; tÞdAdðxxÞ dx À LZÀb4 pðx; tÞdAdðxxÞ dx: b3 LÀb4 Àb2 It may be written as (11) dW dV ftðtÞ ¼ ÀmW dt À mV dt ; where mW ¼ rA1 a2 þ a À 22b2 3 þ a2 À 1 ðb3 þ b4 þ 2lÞ ; h i mV ¼ rA1 2 1 À aÀ1 b2 þ 1 À aÀ2 ðb3 þ b4 þ 2lÞ : (12) (13) (14) The preceding equations show that the forces on the piston, fp, and on the tube, ft, depend on the sum b3 þ b4, not on the lengths b3 or b4 separately, a result that is not unexpected. They also show that the expressions of those forces are linear in the accelerations dW/dt and dV/dt, with no dependence on velocities. From the viewpoint of the axial force ft on the tube, mW may be regarded as an inertial mass associated with the tube (and ﬂoater) acceleration dW/dt; the same applies to mV in connection with the acceleration dV/dt of the piston in the frame of reference ﬁxed to the buoy-tube pair. If the whole tube is of uniform inner cross section, i.e. if a ¼ 1, it is simply MV ¼ MW and ft ¼ mV ¼ mW ¼ 0, a situation that was studied in [16]. 2.2. Piston versus hydraulic turbine The original IPS buoy was conceived with a piston, sliding along the acceleration tube, whose relative motion activates a secondary hydraulic ram (or linear pump) that supplies high pressure liquid (water or oil) to a hydraulic circuit [4]. If energy is to be absorbed from large amplitude waves, the excursion of the primary piston is also relatively large which requires a long rod (possibly longer than 20 m), which, when subjected to compression forces, can cause serious buckling problems. An alternative to the piston pump is a pair of hose pumps as in the AquaBuOY [7,8], which avoids compression loads but whose hydraulic circuit working pressure is much lower than what is attainable by piston pumps. A self-rectifying hydraulic turbine located in the narrower part of the tube may be used instead of a piston, although this seems not to have been proposed before. This avoids the problem of limiting the piston excursion. In order to avoid cavitation, the turbine should be deeply submerged. Naturally, the ﬂow through the turbine is far more complex than the (assumed one-dimensional) ﬂow in the tube. However, for the purpose of accounting for the inertia of the ﬂow through the turbine, we may deﬁne an 108 A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114 equivalent tube element of length b1 and an equivalent cross sectional area A1, as shown by the shaded area in Fig. 3. Since the diameter of the turbine is expected to be much smaller than the diameter of the main part of the acceleration tube, the area ratio a2 ¼ A2/A1 should be much larger than unity. Obviously, the expressions derived above for the piston in tube may be applied to this case, the volume ﬂow rate through the turbine being Q(t) ¼ A1V(t); the turbine pressure head is Dp ¼ fp(t)/A1, where fp is given by Eq. (8). 2.3. Hydrodynamic analysis in regular waves We consider now the IPS buoy, and denote by x(t) the coordinate for the heaving motions of the ﬂoater-tube set, with x ¼ 0 in the absence of waves and x increasing upwards. Let y(t) be the oscillations in piston position relative to the buoy-tube set. We note that it is x_ ¼ W and y_ ¼ V, where W is the tube velocity and V is the relative velocity of the piston as deﬁned in Section 2.1. We consider a linear PTO such that a relationship fp ¼ Ky þ Cy_ (15) holds between the force on the piston fp and the relative displacement y and velocity y_ of the piston. The constant C is the PTO damping coefﬁcient and the constant K may be regarded as a spring stiffness. The instantaneous power absorbed by the PTO is P ¼ fpy_ . The following hydrodynamic analysis is based on linear water wave theory, which, as is well known, requires the wave amplitude and the amplitude of body oscillations to be small (compared with wavelength). The equation of motion can be found in [17]. We consider the body to consist of a ﬂoater (subscript 1a) and a tube (subscript 1b), and denote by m1a, m1b and m1a, m1b the corresponding masses and added masses. The added mass m1b of the tube is supposed to be independent from the wave frequency (as in an unbounded medium). Provided that the PTO is linear (as assumed above) and after the transients related to the initial conditions have died out, we may write, for the motion of the body in the presence of incoming sinusoidal waves of frequency u, buoy ðM1a þ M1bÞ€x þ Bx_ þ rgSx ¼ feðtÞ þ ftðtÞ þ fpðtÞ: (16) Here M1a ¼ m1a þ m1a and M1b ¼ m1b þ m1b are the mass plus added mass of bodies 1a and 1b, respectively, r is water density, g is acceleration of gravity, B(u) is radiation damping coefﬁcient of the buoy (body 1a), and S is the cross sectional area of the ﬂoater deﬁned by the undisturbed free surface. We note that m1a is a function of u and recall that it is x_ ¼ W where W is the tube velocity deﬁned in Section 2.1. On the right-hand-side of Eq. (16), fe is the hydrodynamic excitation force on the ﬂoater due to the incoming waves, ft is the force on the inner surface of the tube given by Eq. (12) and fp is the vertical force on the piston (given by Eq. (8)) that is transmitted to the buoy by the piston rod or by the pair of hose pumps. We recall that negligible hydrodynamic interference is assumed. We note also that m1b is supposed not to be a function of frequency u (as a consequence from the assumption of deep submergence of body 1b). Since we have a linear system acted upon by a simple-time- harmonic excitation force of frequency u, we may write, after the transients related to the initial conditions have died out, fx; y; V ; feg ¼ fX; Y; V0; Fegeiut: (17) Here X, Y, V0 ¼ iuY and Fe are complex amplitudes. We may write Fe ¼ AwG(u), where Aw is the incident wave (real) amplitude, and G is the (in general complex) excitation force coefﬁcient. The absolute value of G(u) may be related to B(u) by the Haskind relation (valid for an axisymmetric body oscillating in heave in deep water, see [17]) jGðuÞj ¼ 2g3urB3ðuÞ1=2: (18) By using the complex amplitude representation, we easily obtain, from Eqs (8), (12), (16) and (17), À u2ðM1a þ M1b þ mW þ MWÞX þ iuBX þ rgSX À u2ðmV þ MVÞY ¼ Fe; (19) u2MWX þ u2MVY ¼ ðK þ iuCÞY: (20) For given wave frequency u and excitation force amplitude Fe, the pair of linear algebraic equations (19) and (20) yield the complex amplitudes X and Y. The time-averaged value of the power absorbed by the PTO (piston or turbine) is P ¼ u2CjYj2=2. If the whole tube is of uniform inner cross section, i.e. if a ¼ 1, it is simply A1 ¼ A2, ft ¼ mV ¼ mW ¼ 0 and MV ¼ MW ¼ M2 (say), where M2 is the mass plus added mass of the water contained in the tube. In this case, Eqs (19) and (20) reduce to turbine Fig. 3. IPS buoy with the piston replaced by a hydraulic turbine (in the shaded space). Àu2ðM1a þ M1b þ M2ÞX þ iuBX þ rgSX À u2M2Y ¼ Fe; (21) u2M2X þ u2M2Y ¼ ðK þ iuCÞY: (22) Since we only consider heave oscillations, the equations of motion are not affected by how the mass m1 ¼ m1a þ m1b is distributed between bodies 1a and 1b. For convenience of presen- tation of numerical results, we assume that m1a is the mass of water of volume equal to the submerged part of the buoy in the absence of waves. The mass m1b and the added mass m1b of body 1b appear together in the equations as M1b ¼ m1b þ m1b. For this reason, the numerical results in Sections 3 and 4 are given for M1b and not for m1b and m1b separately. A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114 109 3. Numerical results in regular waves Numerical results were obtained for a cylindrical buoy of radius a, with a conical bottom (semi-angle of the cone equal to p/3). In calm water, the cylindrical part of the buoy is submerged to a depth equal to the radius a. The added mass m1a and the radiation damping coefﬁcient B were computed with the software WAMIT for a set of values of the frequency u and deep water. A dimensionless plot of m1*a ¼ m1a/(rpa3) and B* ¼ B/(rpa3u) versus T* ¼ T(g/a)1/2 (T ¼ wave period) is shown in Fig. 4. We deﬁne the dimensionless value of the mass of body 1 (including the added mass of body 1b) as M1* ¼ (m1a þ M1b)/m1a, where, for the geometry considered here, it is m1a ¼ rp a3(1 þ 3À1tan p/6) ¼ 3.7462r a3. We note that the minimum value of M1* is equal to unity, since M1b ¼ m1b þ m1b cannot be negative. In the special case when a ¼ 1, i.e. an acceler- ation tube of uniform inner cross section (A1 ¼ A2), we also deﬁne the dimensionless mass plus added mass of body 2 (water con- tained in the tube) as M2* ¼ M2/m1a, where M2 ¼ r A1(L þ 2l). Here, we consider regular waves of frequency u and assume that the PTO consists solely of a linear damper (no spring, i.e. K ¼ 0). We deﬁne a dimensionless damping coefﬁcient C*(u) ¼ C/B(u), where B(u) is the radiation damping coefﬁcient of body 1a. We also deﬁne dimensionless values X* ¼ jXj/Aw (Aw ¼ incident wave amplitude) for the motion amplitude of body 1 (buoy-tube set) and Y* ¼ jY/Xj for the amplitude of the piston motion relative to the buoy-tube pair. Note that Y* ¼ 0 means that the piston is rigidly connected to the buoy. If the piston does not move (possibly because the inertia of the water inside the tube is inﬁnite) it is x_ ¼ Ày_ and Y* ¼ 1. The theoretical maximum limit for the time-averaged wave power that can be absorbed from regular waves in deep water by a heaving wave energy converter with a vertical axis of symmetry is well known to be (see [17]) Pmax ¼ g3rA2w 4u3 : (23) Accordingly we deﬁne the dimensionless power P* ¼ P=Pmax 1, where P ¼ u2CjYj2=2 is the time-averaged power absorbed from the waves. 3.1. Tube of uniform cross section a ¼ 1 We consider ﬁrst the case when a ¼ 1 (and ft ¼ 0). Since the inner cross section of the tube is uniform, the ﬂow of water (assumed one-dimensional) inside the acceleration tube is also uniform (the water moves as a solid body) and the system is equivalent to a two-body heaving wave energy converter in which the mass plus added mass of bodies 1 and 2 are respectively M1a þ M1b and M2, and the PTO is activated by the relative motion between the bodies. This two-body case was theoretically analysed in detail by Falnes [18]. An optimization was performed that consisted in ﬁnding the pair of dimensionless values C* and M2* that maximizes P*, for given values of the dimensionless wave period T* and of M1*. This twodimensional optimization was performed with the aid of the FindMaximum subroutine of Mathematica. Results are shown in Figs. 5e7 for T* ¼ 10, 12 and 14. The following curves (dimensionless values) are plotted (versus M1*): (i) amplitude Y* ¼ jY/Xj of the relative motion between bodies 1 and 2; (ii) M2* (mass plus added mass of body 2); (iii) PTO damping coefﬁcient C*. For all plotted points, it is P* ¼ P=Pmax ¼ 1, since maximum capture width l/2p (l ¼ wavelength) is attained by the maximization process. We note that the wave energy is absorbed solely from the motion of body 1a (bodies 1b and 2 are assumed far away from the free surface). So X* is the same as for a single body 1a optimally reacting against the bottom; it depends only on T* and is independent of the optimal pair (M1*, M2*). It is well known (see e.g. [17]) that, for a single heaving body, maximum absorbed power is attained for oscillation amplitude Xopt ¼ jFej(2uB)À1, which, for axisymmetric ﬂoating body 1a, can be written in dimensionless form as Xo*pt ¼ ð2pÞÀ7=2B*À1=2T*3; (24) 1.2 1 0.8 0.6 M * 2 0.4 Y* 0.2 0 1 1.2 1.4 1.6 1.8 40 30 20 C* 10 Fig. 4. Dimensionless plot of the added mass m1*a (solid line) and radiation damping coefﬁcient B* (dotted line) versus wave period T*, for the buoy (body 1a) in deep water. 0 1 1.2 M1* 1.4 1.6 1.8 Fig. 5. Dimensionless plots of M2*, cross section) and wave period T* Y* ¼ and C* versus 10. Maximum M1*, for a ¼ 1 (tube absorbed power P of uniform inside * ¼ 1 is attained for all plotted points. 110 A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114 2.5 3 2 1.5 1 M * 2 0.5 Y* 0 1 1.25 1.5 1.75 2 2.25 2.5 2.75 175 150 125 100 75 C* 50 25 0 1 1.25 1.5 1.75 2 2.25 2.5 2.75 M1* Fig. 6. As in Fig. 5, for T* ¼ 12. where B* ¼ B(r pa3u)À1. We recall that B* is a function of T*. The optimal value Xo*pt, for buoy 1a, is plotted versus T* in Fig. 8 and can be seen to increase rapidly with T* in the plotted range. In Figs. 5e7, the curve for M2* exhibits a minimum, for a value of M1* that increases with T*. To the right of this point, M2* increases to inﬁnity. In the limiting situation when M2* ¼ N, we have what is equivalent to a single body (1a þ 1b) reacting against the bottom (Y* ¼ 1), for which case the optimal conditions are well known (see e.g. [17]): M1a þ M1b ¼ r gS/u2 and C ¼ B (i.e. C* ¼ 1). In this limiting situation, it is M1**b ¼ M1b m1a ¼ rgS u2m1a À m1a m1a À 1; (25) which, for the special case of the cylindrical buoy with conical bottom considered here, becomes M1**b ¼ 0.02124T*2 À 0.8387m1a/ m1a À 1. (Note that, like B1*a, m1a/m1a is also function of T*.) This equation is plotted in Fig. 9. It is M1**b ¼ 0 for T* ¼ 7.940, which means that body 1a, if isolated, would be perfectly tuned to this wave period. We note that Y* increases and C* decreases with increasing M1*, with (as should be expected) Y* / 1 and C* / 1 as the limiting case M2* ¼ N is approached. Since, for the plotted points, it is P* ¼ 1, it can easily be shown that, for ﬁxed T* and varying M1*, C* is proportional to Y*À2. The dimensionless amplitude of the PTO force oscillations C*½Y*À1 is proportional to C* (or to Y*) and, as seen in Figs. 5e7, increases rapidly with decreasing M1*. 3.2. Tube of non-uniform cross section a > 1 If a > 1, it is A2 > A1, and the ﬂow in the acceleration tube is no longer uniform. For this reason, the presence and inertia of water 2.5 2 1.5 M * 2 1 0.5 Y* 0 1 1.5 2 2.5 3 3.5 4 500 400 300 200 C* 100 0 1 1.5 2 2.5 3 3.5 4 M1* Fig. 7. As in Fig. 5, for T* ¼ 14. inside the tube can no longer be represented simply by a solid body and its mass. Besides, the force ft on the conical parts of the tube inner surface is non-zero and may be signiﬁcant. So (unlike if a ¼ 1), more than one parameter is now required to describe the water contained in the tube and the forces associated with it. In addition to a, we use, as parameters, the tube diameter D2 ¼ 2(A2/p)1/2, the length b1 of the central part of the tube, the total tube length L, and the half-angle b ¼ arctan D2 2b2 1 À aÀ1 (26) 20 15 Xo∗pt 10 5 0 8 10 12 14 16 T∗ Fig. 8. Dimensionless plot of optimal oscillation amplitude Xo*pt of buoy 1a versus wave period T*. A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114 111 5 4 L* 3 2 1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Fig. 9. Plot of M1**b ¼ 0.02124T*2 À 0.8387m1a/m1a À 1 for the considered cylindrical buoy with conical bottom. of the inner conical walls of the tube. As before, we introduce dimensionless quantities D2* ¼ D2/a, b1* ¼ b1/a and L* ¼ L/a (where a is the ﬂoater radius). It is D2* ¼ D2/a ¼ 0.8 for all the plotted results. Since the length b3 þ b4 of the tube segments of diameter D2 should be non-negative, the following restriction applies L* ! b*1 þ D*2 1 À aÀ1 cot b: (27) Fig. 10 shows results for T* ¼ 10. A geometry representative of an IPS buoy with piston was chosen: b1* ¼ 0.533, b ¼ p/6 and a ¼ D2/ D1 ¼1.25. Results for a ¼ 1 are also shown for comparison. For all plotted points, the maximization procedure yielded P* ¼ 1. It can be seen that, for ﬁxed T* and M1*, a larger value of a (1.25 as compared with 1) results in larger tube length L*, larger piston displacement amplitude Y* and smaller PTO damping coefﬁcient. If a hydraulic turbine is to be used instead of a piston, a much larger value of a should be chosen as well as a much smaller value of b1*. Fig. 11 shows results for a ¼ 4, D2* ¼ 0.8, b1* ¼ 0.2 and b ¼ p/6. Two wave periods are represented: T* ¼ 10 and 12. As before, the maximization procedure yielded P* ¼ 1 for all plotted points. The differences with respect to the case a ¼ 1 (tube of uniform inner cross section) are now much more marked, especially on what concerns Y* and C*. It is interesting to examine in detail the curves L* versus M1* plotted in Fig. 11 for a ¼ 4 and compare them with the curves of M2* versus M1* plotted in Figs. 5 and 6 for a ¼ 1. We recall that it is P* ¼ 1 for all plotted points in these ﬁgures. In all cases, the curves exhibit a minimum. But, while in Figs. 6 and 7 the curves to the right of this point rise to M2* ¼ N (in the limit representing the single body 1a þ 1b reacting against the sea bottom), in Fig. 11 the curves are of ﬁnite extent: the condition P* ¼ 1 cannot be maintained beyond the last plotted point, well short of L* ¼ N. In what follows we attempt to explain this different behaviour. Let us denote by V1 the relative ﬂow velocity, and by U1 ¼ V1 þ W the absolute ﬂow velocity, in the central part of the tube where the cross sectional area is A1 (we recall that W is the velocity of the buoy-tube set). Then, in the part of the tube where the area is A2 ¼ a2A1, the relative ﬂow velocity is V2 ¼ aÀ2V1 and the absolute ﬂow velocity is aÀ2V1 þ W ¼ U2 (say). If we ﬁx the length b1 of the central part of the tube and let b3 þ b4 / N (i.e. L / N), then, because of the inertia of the very large volume of enclosed water, it will be U2 / 0 and, consequently, V1 / Àa2W and U1 / ÀW(a2 À 1). We may say that, for a > 1, even if the tube length becomes very large, the absolute velocity of the piston (or of the ﬂow admitted to the hydraulic turbine) does not vanish and its direction is opposite to the velocity direction of the buoy-tube pair. This, together with the fact that the force ft on the 1.5 1.25 Y* 1 0.75 0.5 0.25 0 1 40 1.1 1.2 1.3 1.4 1.5 1.6 1.7 30 C* 20 10 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 M1* Fig. 10. Dimensionless plots of L*, Y* and C* versus M1*, for a ¼ 1 (open symbols) and a ¼ 1.25 (closed symbols). Tube geometry: b1* ¼ 0.533, b ¼ p/6. Wave period T* ¼ 10. Maximum absorbed power P* ¼ 1 is attained for all plotted points. inner surface of the tube is non-zero, explains why, if a > 1, from the viewpoint of device performance, the water contained in a very long acceleration tube (b3 þ b4 / N) is not equivalent to a solid body of inﬁnite mass. It should be noted that, especially in Fig. 10, for some plotted points, the dimensionless tube length L* ¼ L/a ¼ 1.25L/D2 is relatively small (in some cases less than 2) which makes the assumption of negligible hydrodynamic interference between the tube ends to be a rough approximation. Naturally, in such cases, the numerical values should be regarded as giving qualitative, rather than quantitative, information. Besides, such cases are likely to be of limited practical interest, since the tube length would be insufﬁcient for the required piston excursion. 4. Numerical results in irregular waves Not surprisingly, the results of the optimization (masses, tube length, damping coefﬁcient) in regular waves were found to depend strongly on wave period T. This means that the performance in irregular waves should be expected to be signiﬁcantly 112 A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114 14 12 10 L* 8 6 4 2 0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 7 6 5 Y* 4 3 2 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0.8 0.6 C* 0.4 0.2 0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 M1* Fig. 11. Dimensionless plots of L*, Y* and C* versus M1*. Tube geometry: a ¼ 4, b1* ¼ 0.2, b ¼ p/6. Wave periods T* ¼ 10 (open symbols) and 12 (closed symbols). Maximum absorbed power P* ¼ 1 is attained for all plotted points. inferior to what can be achieved in regular waves, since the irreg- ular waves may be modelled as a superposition (spectral distribu- tion) of regular waves, and the device (equipped with a linear damper) cannot be optimally tuned simultaneously to all freq- uencies of the spectrum. To investigate this, a similar (but more realistic) optimization procedure was performed along the same lines, for irregular waves. This consisted in ﬁnding the pair of values of tube length L and PTO damping coefﬁcient C that, for each value of the mass M1b, maximizes the time-averaged absorbed power. A PiersoneMoskowitz spectral distribution was adopted, deﬁned by (SI units, see [19]) SðuÞ ¼ 526Hs2TeÀ4uÀ5exp À 1054TeÀ4uÀ4 ; (28) where Hs is signiﬁcant wave height and Te is energy period. The time-averaged power output in irregular waves is computed as ZN PirrðHs; TeÞ ¼ P1ðuÞSðuÞ du; 0 (29) Fig. 12. Dimensionless plot of time-averaged power P*irr versus M1*, from optimization in irregular waves, for a ¼ 1, 1.25 and 4, and Te* ¼ 8 and 10. where P1ðuÞ is the time-averaged power absorbed from regular waves of frequency u and unit amplitude. Results are shown in Figs. 12e14 for Te* ¼ Te(g/a)1/2 ¼ 8 and 10. As for regular waves, the following geometries were adopted: D2* ¼ D2/ a a ¼ ¼ 0.8, 4). b ¼ p/6, b1* Curves are ¼ b1/a given ¼ 0.533 in Fig. (for a 12 for ¼1 P*irr and 1.25) and ¼ Pirr=Pmax;irr 0.2 (for versus M1*b ¼ M1b/m1a, where Pmax;irr ¼ g3r ZN uÀ3SðuÞ du 4 ¼ 149:5Hs2 Te3 0 (30) (SI units) is the maximum power that can be extracted by an axisymmetric body in deep water oscillating in heave from a sea state represented by the spectral distribution S(u). Curves are also shown for the dimensionless optimal length ratio L* ¼ L/a (Fig. 13) and dimensionless damping ratio C* ¼ C/B(2p/Te) (Fig. 14) (as before, B(u) is the hydrodynamic radiation damping coefﬁcient of body 1a). Fig. 12 shows that (for M1* > 1) the maximum value of P * irr does not exceed 0.38 (if Te* ¼ 8) or 0.31 (if Te* ¼ 10). Unlike in regular waves, we are now far from attaining the theoretical maximum. The curves for a ¼ 1 and 1.25 (values typical of the original IPS concept) are practically coincident. This means that the introduc- tion, in the acceleration tube, of a working section whose diameter D1 is not much smaller than D2 does not affect signiﬁcantly the capability of energy extraction from the waves. The same is not true Fig. 13. Dimensionless plot of tube length L* versus M1*, from optimization in irregular waves, for a ¼ 1, 1.25 and 4, and Te* ¼ 8 and 10. A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114 113 5. Conclusions Fig. 14. Dimensionless plot of PTO damping coefﬁcient C* versus M1*, from optimization in irregular waves, for a ¼ 1, 1.25 and 4, and Te* ¼ 8 and 10. if a is much larger than unity, as shown by the curve for a ¼ 4 (representative of a PTO using a hydraulic turbine, rather than a piston, inside the tube) that signiﬁcantly departs from the other two curves (especially for Te* ¼ 8). For a ¼ 1 and 1.25, the curves for optimal L* (as in regular waves) show a minimum, that separates the “strong damping” from the “weak damping” solutions. This is no longer the case (or is much less marked) for a ¼ 4. The optimal value of the PTO damping coefﬁcient C* decreases monotonically with increasing M1*, as shown in Fig. 14 (and as was found above for regular waves in Section 3). That value also decreases rapidly with increasing a (more marked nozzle effect) as expected since the ﬂow velocity in the central section of the tube (of area A1, where the piston or the hydraulic turbine is located) increases with the cross sectional area ratio A2/A1 ¼ a2. It appears that the use of a turbine (large value of a) instead of a piston (a close to unity), although possibly having some practical advantages, is likely to result in a poorer wave energy absorption and require a longer tube. We note that these optimized results assume that the mass M1b and tube length L (in addition to the PTO damping coefﬁcient) may be changed to match the sea state (especially the energy period Te), which is likely to be highly unpractical. Even if this were done, the results for captured energy are found to be relatively poor, especially for the larger energy periods. The poor performance in irregular waves (as compared with regular waves) should be associated to the limitations of a PTO that provides only pure (linear) damping. In fact, it is known that optimal reactive control would allow the theoretical maximum to be attained (i.e. P*irr ¼ 1). Unfortunately optimal reactive control (apart from requiring a PTO difﬁcult to realize) is hardly attainable, for well known reasons [17,20]: it would require the prediction of the incoming waves (and also relatively heavy computing that cannot be easily implemented in real time). A frequency domain analysis of the IPS hydrodynamics, combined with a one-dimensional model of the (in general nonuniform) unsteady ﬂow inside the acceleration tube, has been developed and used to assess the performance of the device in regular incident waves. In spite of somewhat drastic simplifying assumptions concerning the wave ﬁeld interference between buoy and tube, the obtained results are believed to be signiﬁcant. If the cross section of the acceleration tube is non-uniform (which could be dictated by practical reasons, namely to limit the piston excursion or to allow a hydraulic turbine to be installed), the ﬂow inside the tube is also non-uniform, and the inertia of the enclosed water cannot be represented by that of a solid body. Besides, apart from the axial force on the piston (or on the hydraulic turbine), the extra axial force on the non-cylindrical inner walls of the tube has to be accounted for. In such situations, the dynamics of the IPS buoy can no longer be theoretically modelled as Falnes [18] did for a two-body heaving system or as done in [16] for the IPS buoy. In regular waves, it was found that, for a given buoy and given tube diameter D2 and diameter ratio a, maximum wave energy absorption (equal to what can be achieved theoretically by an axisymmetric heaving body) is attained by an inﬁnite number of combinations of mass M1b, tube length L and PTO damping coefﬁ- cient C. If D2 and M1b are kept ﬁxed and a allowed to increase above unity (or D1 to decrease below D2, i.e. a more marked nozzle effect), the conditions for maximum energy absorption require a longer tube, a longer piston excursion and a smaller damping coefﬁcient. In irregular waves, the performance was found to be much poorer (as should be expected since no phase control is considered), even if the mass M1b, the tube length L and the PTO damping coefﬁcient C are optimized. The use of a hydraulic turbine in the tube instead of a piston was found in general to result in poorer wave energy absorption and to require a longer tube. In any case (regular or irregular waves), for given ﬂoater and wave period, the designer of the acceleration tube and PTO is free to choose among a wide range of values for M1b, L and C without signiﬁcantly impairing the device power performance. The best criterion, from the engineering viewpoint, is not necessarily the minimum value of the tube length L, since this could require an excessive value of the buoy-plus-tube mass (and volume). It should be pointed out that the choice of the pair of optimal values of M1b and L may result in widely different values of the PTO optimal damping coefﬁcient C (and of the piston excursion) which may signiﬁcantly affect the PTO design. All the presented numerical values are for a ﬁxed ratio D2/ a ¼ 0.8. It seems reasonable to assume that a longer tube would be required for smaller tube diameter D2. The theoretical analysis presented here is based on several simplifying assumptions, some of which are possibly not satisﬁed by some geometries of practical interest (namely on what concerns the tube length and the distance from the upper end of the tube to the free surface). In such cases the results should be regarded as qualitative rather than quantitative and the present analysis complemented by a more rigorous numerical approach possibly based on the boundary element method. Acknowledgements The work reported here was partly supported by the Portuguese Foundation for Science and Technology through funding to IDMEC/LAETA and under contract PTDC/EME-MFE/103524/2008. Co-author JCCH thanks Programa Ciência 2007 for ﬁnancial support. 114 A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114 References [1] Falcão AF de O. Wave energy utilization: a review of the technologies. Renew Sust Energy Rev 2010;14:899e918. [2] Masuda Y. Wave-activated generator. Int. colloq. exposition oceans, Bordeaux, France; 1971. [3] Masuda Y. An experience of wave power generator through tests and improvement. In: Evans DV, Falcão AF de O, editors. Hydrodynamics of ocean wave-energy utilization. Berlin: Springer-Verlag; 1986. p. 445e52. [4] Noren SA. 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